Traveling Waves for Conservation Laws with Nonlocal Flux for Traffic Flow on Rough Roads

We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $x=0$. We study stationary traveling wave profiles cross $x=0$, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asymptotic solutions for the Cauchy problem of the conservation laws under mild assumption on the initial data, while other profiles are unstable.

The function κ(x) denotes the speed limit of the road at x, which represents the road condition. We consider rough road condition where κ(x) can be discontinuous.
The models (M1) and (M2) can be formally derived as the continuum limit of particle models where the speed of each car follows certain rules. For model (M1), the speed of the car at x is where a weighted average of car density on an interval of length h in front of the driver (i.e., downstream) is computed. For model (M2), the speed of the car at x is x+h x κ(y)v(ρ(t, y))w(y − x) dy, where the weighted average is taken over κ(y)v(ρ(t, y)) over an interval of length h in front of the driver. The weight w(x) is a non-negative function defined on x ≥ 0. For a given h, we assume that w(x) is continuous and bounded on x > 0 with its support on [0, h), and satisfies h 0 w(x) dx = 1, w(x) = 0 ∀x ≥ h, w (x) < 0 ∀x ∈ (0, h). (1.4) Here, the assumption w (x) < 0 indicates that the condition right in front of the driver is more important than those further ahead. Formally, as h → 0+ and w(·) converges to a dirac delta, both (M1) and (M2) converge to a scalar conservation law with local flux Unfortunately rigorous proofs of such convergences have not been established yet in the literature, not even for the case where κ(x) is a constant function. Non-local conservation laws has gained growing interests in recent years. In the simpler case where κ(x) ≡ 1, the existence and well-posedness of solutions of the Cauchy problem for (M1) were established in [6], using numerical approximations generated by the Lax-Friedrich scheme. The same results for (M2) were proved in [12], using numerical approximations generated by a Godunov-type scheme. Other models of conservation laws with nonlocal flux include models for slow erosion of granular flow [3,18], for synchronization [2], for sedimentation [5], for nonlocal crowd dynamics [8], and for material with fading memory [7]. Models with symmetric kernel functions have been studied in [19], and for systems in several space dimensions [1]. Some numerical integrations are studied in [4], and an overview of several nonlinear nonlocal models can be found in [11].
In this paper we are interested in the traveling wave profiles for (M1) and (M2). We remark that, when the road condition is uniform, say κ(x) ≡ 1, the traveling wave profiles for (M1) and (M2) are studied in a recent work [15], where we established various results on existence, uniqueness and stability of the traveling wave profiles. In this paper, however, we consider rough road condition, where κ(x) is discontinuous. To fix the idea, we consider κ(x) = κ − , x < 0, κ + , x > 0. (1.6) The main objective of this paper is to study the stationary traveling wave profiles of (M1) and (M2), crossing the jump in κ(x) at x = 0. For all possible cases, we show analytical and numerical results on existence (and non-existence), uniqueness (and non-uniqueness) and stability (and instability) of these traveling wave profiles. Traveling wave profiles for a local follow-the-leader model for traffic flow was studied in [17] for homogeneous road conditions, and in [16] with rough road condition. We also mention that, for the non-local models for slow erosion of granular flow, traveling waves and their local stability were studied in [13]. The rest of the paper is organized as follows. In Section 2 we consider model (M1) and analyze two cases with κ − > κ + and κ − < κ + , where each case has 4 sub-cases. In Section 3 we analyze model (M2). Final remarks are given in Section 4.

Stationary traveling wave profiles for (M1)
We seek a stationary profile Q(x) for (M1) around x = 0. To simplify the notation, we introduce an averaging operator Note that, since w(x) vanishes outside the interval (0, h), we could put the upper integration bound to be ∞. Furthermore, as long as In the case when the following constraint on ρ − , ρ + must be imposed Differentiating (2.1) in x, we obtain an delay integro-differential equation Here, δ 0 (x) denote the dirac delta. Since κ(x) and Q(x) may be discontinuous, in the lefthand side of the equation the quantities are evaluated at (x−). For general theory on delay differential equations we refer to [9,10]. The profiles Q(x) are discontinuous at x = 0. Since A(Q; x) is continuous at x = 0, so must κ(x)Q(x). We impose the connecting condition on the traces

Review of previous results
The simpler case where κ(x) ≡κ = constant is studied in a recent work [15]. We review some related results which will be useful in the analysis of this paper. Under the assumption (1.3), there exists a unique stagnation pointρ where Let W (x) be a stationary profile for (M1) with κ(x) ≡κ. It satisfies the following integroequation Next Lemma was proved in [15] (Lemma 3.1).
Letρ be the stagnation point satisfying (2.7). The following holds.
The following existence and uniqueness result of the profile is proved in Theorem 3.2 of [15].
There exist solutions of W (x) which are monotone increasing. Furthermore, the solution is unique up to horizontal shifts.
In the remaining of this section we let W (x) denote the smooth and monotone profile with κ(x) ≡ κ + and the boundary conditions This implies that ρ + ≤ρ is an unstable asymptote for x → ∞, and ρ − ≥ρ is an unstable asymptote for x → −∞.
When κ(x) is discontinuous, the profiles Q(x) are very differently. We discuss two cases in the next two sections, where Case A is for κ − > κ + , and Case B for κ − < κ + . For notational convenience, we introduce the functions:
For the rest of the paper we only consider the nontrivial case withf > 0, where 0 < ρ ± < 1.
Proof. The proof takes several steps.
Step 1. Since ρ + >ρ is a stable asymptote as x → ∞, then on x > 0, the profile Q(x) can be either the constant function Q(x) ≡ ρ + , or some horizontal shifts of W (x). This solution is continuous and monotone increasing on x > 0.
Step 2. At x = 0 the profile Q(x) has an upward jump. The traces Q(0−) and Q(0+) satisfy Furthermore, recalling the definition of ρ 2 in (2.11), we have Then it holds Step 3. With this given Q(x) on x ≥ 0, we solve an "initial value problem" for Q(x) backward in x for x < 0. In order to establish the existence of solutions for the initial value problem, we generate approximate solutions by discretization. Fix the mesh size ∆x, we have the discretization On x < 0, the approximate solution Q ∆x (x) is reconstructed as the linear interpolation through the discrete values Q i , i.e., The discrete values Q i can be generated iteratively. Given a profile Q ∆x (x) on x ≥ x i , we compute the value Q i−1 by solving the nonlinear equation Step 4. We now verify that (2.14) has a unique solution. We compute: Furthermore, for ∆x sufficiently small, we have Thus, (2.14) has a unique solution of Q i−1 , satisfying 0 < Q i−1 < Q i . We remark that, numerically, (2.14) can be solved efficiently by Newton iterations, using Q i as the initial guess.
By induction we conclude that 0 < Q i−1 < Q i for all i < 0. Thus the approximate solution Q ∆x (x) is positive and monotone increasing on x < 0. By construction, it satisfies the equation (2.1) at every grid point x i . Taking the limit ∆x → 0, Q ∆x (x) converges to a limit function Q(x), positive and monotone increasing, and satisfies (2.1) for every x.
Step 5. It remains to verify the limit as x → −∞. Since Q(x) is monotone and bounded, it must admit a limit at x → −∞. Call it ρ − . We havē We must also have ρ − <ρ, since by Lemma 2.1 it is the only stable asymptote as x → −∞. This completes the proof. Several sample profiles are plotted in Figure 2. They are generated by numerical simulation, solving (2.14) by Newton iterations. All profiles are bounded and monotone, continuous except an upward jump at x = 0. We further note that the profiles do not cross each other. In other words, the profiles are ordered. We have the following Corollary.
Corollary 2.1 (Ordering of the profiles). Consider the setting of Theorem 2.2, and let Q 1 (x) and Q 2 (x) be two distinct profiles. Then, we must have either Proof. First we observe that on x > 0, the profiles can not cross each other, since they are horizontal shifts of the monotone profile W (x). Now consider two distinct profiles Q 1 , Q 2 , such that Q 1 (x) > Q 2 (x) on x > 0. Then Q 1 (0+) > Q 2 (0+), and by the connecting condition (2.5), we have also Q 1 (0−) > Q 2 (0−). We continue by contradiction, and assume that there exists a point y < 0 such that Then, by equation (2.1) we have a contradiction to (2.16). We conclude that the graphs of Q 1 , Q 2 does not cross each other.
Stability. We now show the stability of these profiles, as local attractors for solutions of the Cauchy problems for (M1) with suitable initial data. When κ(x) ≡ 1 is a constant function, the existence and uniqueness of solutions for the Cauchy problems of (1.1) is established in [6].
is discontinuous at x = 0, then the solution ρ(t, x) must have a jump at x = 0 as well. A connecting condition, similar to (2.5), should be imposed: We have the following definition.
Definition 2.1. We say that a function ρ(t, x) is a solution of (1.1) if ρ(t, x) satisfies (1.1) for all x > 0 and x < 0, and the connecting condition (2.17) at x = 0, for all t > 0.
We remark that, a general existence theorem for the Cauchy problem of (M1) with discontinu- is not yet available in the literature. One may speculate that the existence of solutions could be established by the vanishing viscosity approach, a possible topic for future work. In this paper, assuming that solutions exist for the Cauchy problem, we establish the local stability of the traveling wave profiles. Let Q (x) and Q (x) be two profiles such that Q (x) > Q (x) for every x. Define D as the region between Q and Q , i.e., By Corollary 2.1, all these profiles are ordered and they do not intersect with each other. One can parametrize each profile by its trace Q(0+). For every point (x, q) ∈ D with x = 0, there is a unique profile that passes through it. Fix a time t ≥ 0. For each function ρ(t, x) with (x, ρ(t, x)) ∈ D and x = 0, we define the mapping: We have the following stability Theorem. Letρ(x) be the initial data, smooth except at x = 0, satisfying Let ρ(t, x) be the solution of the Cauchy problem for (M1) with initial dataρ(x), following Definition 2.1. Then, we have Proof. We first observe that, since the initial data is smooth except at x = 0, so is the solution for t > 0. We now claim the following: This claim would imply both (2.21) and (2.22). We provide a proof for the case of a maximum point, while the minimum point can be treated in a completely similar way. Fix a time t ≥ 0, and let y be a point that satisfies (2.23). We discuss 3 cases, for different locations of y.
The rest of the computation remains the same as in the previous steps.
We perform a numerical simulation with Riemann initial data (ρ − , ρ + ), and the plots are shown in Figure 3. We observe that, the solution approaches a traveling wave profile as t grows.
Note that the Riemann initial data actually do not satisfy the assumptions in Theorem 2.3, indicating that Theorem 2.3 probably applies to more general functions of initial data.

Case A2: ρ − < ρ + ≤ρ
Since ρ + ≤ρ is an unstable asymptote for x → ∞, the solution on x > 0 must be Q(x) ≡ ρ + . At x = 0 the connecting condition (2.5) implies Q(0−) < Q(0+), therefore the profile has an upward jump. Finally, the solution can be extended to x < 0 by solving an initial value problem for (2.1), with initial data given on x > 0, and a jump at x = 0. Using the same argument as in Theorem 2.2 for Case A1, we conclude that this unique profile is monotone increasing on x < 0. We have the following Theorem.
See Figure 4 for a sample profile. Since ρ + ≤ρ is an unstable asymptote for x → ∞, the profile is not a local attractor for solutions of the Cauchy problems for (M1). In fact, any perturbation that enters the region x > 0 will persist, as verified by a numerical simulation in Figure 5. We observe that, with Riemann initial data, an oscillation is formed around the origin and then travels into the region x > 0, where it travels further to the right as t grows.

2.2.3
Case A3:ρ ≤ ρ + < ρ − Since ρ − >ρ, the asymptote ρ − is unstable as x → −∞. If a profile shall exists, it must be constant ρ − on x < 0. There exists no profile on x > 0 that can be connected to this constant solution. In conclusion, no stationary profiles exist for this case. A numerical simulation with Riemann initial data is performed, and results are plotted in Figure 6. We observe that oscillations are formed around x = 0, which travel to the left in the region x < 0, where they persist as t grows.

Case A4
: ρ + <ρ < ρ − Since both ρ − and ρ + are unstable, there does not exist any profiles. We present a numerical simulation with Riemann data, and plot the results in Figure 7. We observe rather wild behavior. Oscillations are formed around x = 0, and then enter both regions of x > 0 and x < 0, and they persist as t grows.
Note that since κ − < κ + , the connecting condition (2.5) implies that Q(0−) > Q(0+), thus Q(x) has a downward jump at x = 0. This means that the profiles are no longer monotone increasing. Furthermore, since we seek profiles with Q(0−) ≤ 1, this imposes a restriction on the trace Q(0+), As previously, we let W (x) denote a stationary profile where κ(x) ≡ κ + , discussed in Section 2.1. Below, we discuss each sub-case in detail. We remark that the overall framework of the discussions is similar to that of Case A, but details (especially for Case B1) are rather different, due to the lack of monotonicity.
Proof. The proof takes several steps.
(1) On x > 0, Q(x) could be either the constant function Q(x) ≡ ρ + or some horizontal shift of W (x). We first claim that the constant solution Q(x) ≡ ρ + on x > 0 will not give a profile on x < 0. Indeed, by the connecting condition (2.5) we have Q(0−) > Q(0+) = ρ + . Then one can easily show that Q (0−) < 0, and furthermore Q (x) < 0 for x < 0. Thus, Q(x) reaches 1 for some finite x < 0, and the solution can not be continued further as x becomes smaller.
(2) Furthermore, we consider only the profiles where Q(0+) satisfies (2.28). At x = 0, the connecting condition (2.5) applies, which determines the value for the trace Q(0−). We then solve an initial value problem for Q(x) on x < 0, with initial data given on x ≥ 0. We expect to have infinitely many profiles. From the same argument as in Corollary 2.1, all profiles are ordered and will never cross each other. (4) We now construct a lower envelope Q for all the profiles on x < 0, by solving the initial value problem with initial data Q(x) ≡ ρ 1 < ρ − for x > 0. Note that any horizontal shift of the profile W (x) satisfies W (x) > ρ 1 for x > 0. Since the profiles can not cross each other, we conclude that, any profiles with Q(x) = W (x) on x > 0 will lie above Q . We claim that, on x < 0 the profile Q is monotone decreasing and it lies below ρ − . Indeed, since κ + ρ 1 v(ρ 1 ) = κ − ρ − v(ρ − ) =f , the connecting condition (2.5) gives Then, by (2.4), we have where the last inequality holds thanks to Q(0−) > Q (h) = ρ 1 and we have A(Q ; 0−) x < 0. A contradiction argument shows that (Q ) (x) < 0 for every x < 0. Thus, Q is monotone decreasing on x < 0.
To show that Q (x) < ρ − on x < 0, we proceed with contradiction. We assume that there exists a y < 0 such that Q (y) = ρ − and Q (x) < ρ − for x > y. By (2.1) we compute a contradiction which proofs our claim. Finally, by the same argument as in Step 5 in the proof of Theorem 2.2 one concludes that the profile Q (x) approaches ρ − asymptotically as x → −∞.
(5) By the ordering of the profiles, any profile Q(x) with Q(0−) > ρ 1 , if it exists, would lie above Q (x). However Q(x) might lose monotonicity and become oscillatory around ρ − on x < 0. If this happens, we claim that the local max of an oscillating solution is decreasing and approaches ρ − as x → −∞. Indeed, by (2.1), on x < 0 we have Thus, Q (x) has the same sign as A(Q; x) x . By the assumption w(h) = 0, we have w(0) = − h 0 w (s) ds. Thus, we compute, for x < 0 Since w < 0, we conclude that, if A(Q; x ) x = 0 for some x , then on [x , x + h] we have either Q(x) ≡ Q(x ) or Q(x) oscillated around Q(x ). In particular, this implies that if y < −h is a local max with Q(y) > ρ − , such that A(Q; y) x = 0, then we must have Q(x * ) > Q(y) for some x * ∈ (y, y + h). Therefore, there exists a local max on the left of y, with a higher max value. Thus, there exists a sequence of local maxima y k with The sequence might be finite or infinite. We conclude that there exists an increasing function on x < 0 which serves as an upper envelope for this oscillatory profile. Since the flux equals f , this envelope approaches ρ − asymptotically as x → −∞.
By continuity there exists an upper profile Q (x), whose graph lies between Q (x) and the upper envelope. In particular, Q (x) approaches the limit ρ − as x → −∞.
We conclude that, in between the profiles Q (x) and Q (x), there exist infinitely many profiles for Q(x). These profiles never cross each other, and they all approach ρ − as x → −∞.
Sample profiles for Case B1 are given in Figure 8. By a similar argument as for Case A1, one concludes that these profiles are time asymptotic limits for solutions of the Cauchy problems for (M1). We omit the details of the proof. Result of a numerical simulation is given in Figure 9, with Riemann initial data. We observe that the solution approaches a certain traveling wave profile as t grows.

Case B2, Case B3, and Case B4
For Case B2, we have ρ + < ρ − ≤ρ, which is the counter part for Case A2. Since ρ + ≤ρ is an unstable asymptote as x → ∞, we must have Q(x) ≡ ρ + on x > 0. At x = 0 we apply the connecting condition (2.5) to get Q(0−). We then solve an initial value problem on x < 0. By the same arguments as for Case A2 we prove the existence of a monotone decreasing profile on x < 0. A sample profile is illustrated in Figure 10. Unfortunately, such an profile is not a local attractor for the solutions of the Cauchy problem for (M1). Result of a numerical simulation is given in Figure 11, with Riemann initial data. We observe that an oscillation is formed around x = 0, which travels into the region x > 0, where it persists as t grows.
For case B3, we haveρ ≤ ρ − < ρ + and for Case B4 we have ρ + <ρ < ρ − . These are the counter parts for Case A3 and Case A4 respectively, and there are no profiles. We plot the results of a numerical simulation in Figure 12 for Case B3 and in Figure 13 for Case B4, and observe the oscillations in solutions which persist in time.

Stationary traveling wave profiles for (M2)
Let P (x) be a stationary profile for (M2). It satisfies the equation In the case where lim x→±∞ P (x) = ρ ± and κ(x) satisfies (1.6), we havē Denote now

1) can written as
We see that V (x) is Lipschitz continuous even with discontinuous P (x). This implies that P (x) is also Lipschitz continuous, but with a kink at x = 0. We compute where δ 0 (x) denote the Dirac delta function with unit mass concentrated at x = 0. Writing it out with piecewise details, we have In summary, we seek stationary profiles P (x) which satisfies the equation    Previous results. Consider the simpler case where κ(x) ≡κ is a constant function, and let W(x) be a stationary profile for (M2). Then, W(x) satisfies the integral equation By the results in [15], Lemma 2.1 and Theorem 2.1 in Section 2.1 hold for the profile W(x).
Proof. The proof takes several steps.
Step 1. Since ρ + >ρ is a stable asymptote for x → ∞, then on x ≥ 0 the stationary profile can be either the constant function P (x) ≡ ρ + , or some horizontal shift of the profile W(x). In all cases, the profile is smooth and monotone. With this "initial data" given, the profile P (x) on x < 0 can be obtained by solving an initial value problem of (3.4) backward in x.
Step 2. We now construct an approximate solution to the initial value problem in a similar way as in the proof of Theorem 2.2 (Step 3). We discretize the region x < 0 with a uniform mesh with size ∆x, and denote the grid point by x i = i∆x, for i ∈ Z − . The discrete values P i ≈ P (x i ) are approximate solutions. Using the discrete values, we construct an approximate profile P ∆ (x) as the linear interpolation of the discrete value on x < 0. Given P i for i ≥ k, we construct P k−1 by solving a nonlinear equation where V ∆ is the discrete average velocity Note that we mark the dependence on P k−1 in the corresponding terms. In particular, we write P ∆ (x; Step 3. Assume that P (x) is monotone increasing on x ≥ x k . We claim that, if ∆x is sufficiently small, then the nonlinear equation (3.7) has a unique solution of P k−1 , satisfying 0 < P k−1 < P k . Indeed, we have Furthermore, if we let P k−1 = P k , then P ∆ (x) is constant on the interval x ∈ [x k−1 , x k ], and therefore monotone increasing for x ≥ x k−1 . Then the mapping x → v(P ∆ (x)) is decreasing for x ≥ x k−1 . Since κ − > κ + , then κ(x) is also monotone decreasing. Thus the averages satisfy This gives us and we conclude that there exists a solution of P k−1 satisfying 0 < P k−1 < P k .
In order to show that the solution is unique, we compute the derivative Therefore, for ∆x sufficiently small we have Thus the mapping P k−1 → G is monotone and the root is unique, proving the claim.
Step 4. Set P 0 = W(0). By the above construction, we generate P i for i = −1, −2, · · · , with 0 < P i < P i+1 < P 0 , and thus a positive monotone approximate solution P ∆ (x) on x ≤ 0. By the construction, we have Taking the limit ∆x → 0, the sequence of approximate monotone solutions P ∆ (x) converges to a limit function P (x), which satisfies P (x)V (x) =f for every x ≤ 0. Thus, P (x) is a solution to (3.4), and lim x→−∞ P (x) = ρ − . Finally, for each "initial data" (i.e., some horizontal shift of W(x)) there exists a unit profile P (x), therefore we obtain infinitely many profiles.
Sample profiles are plotted in Figure 14. We observe that they do not cross each other. The ordering property in Corollary 2.1 holds for these profile, with a very similar proof.
Stability. In the case when κ(x) ≡κ is a constant function, the existence and well-posedness of solution for (M2) is established in [12], through convergence of approximate solutions generated by a Godunov-type scheme. Unfortunately, when κ(x) is discontinuous, no existence result is yet available. Assuming that the solutions exist, we show that they converge to the traveling wave profile as t grows, under mild assumptions on the initial data.
Definition 3.1. We say that ρ(t, x) is a solution of (1.2) if ρ(t, x) is continuous and satisfies the equation (1.2) for all x < 0 and x > 0, and the connecting condition With a very similar argument as those for Case A1, one can prove that the profiles P (x) are time asymptotical limit for the Cauchy problem of (1.2). We omit the details of the proof, and state the following stability Theorem.
Theorem 3.2. Let ρ(x, t) be the solution to the Cauchy problem for (1.2) with initial data ρ(x, 0), with κ − > κ + . Assume that the initial data satisfies the connecting condition (3.8) and P (x) ≤ ρ(x, 0) ≤ P (x), for some profiles P and P . Then, the solution ρ(x, t) converges to a profile P (x) as t → ∞.
We perform a numerical simulation with Riemann initial data, and plot the result in Figure 15. We observe that the solution ρ(x, t) quickly approaches a profile, confirming Theorem 3.2. We remark that the initial Riemann data actually does not satisfy the assumptions in Theorem 3.2, and yet we still observe stability. This indicates that the basin of attraction is probably larger than what is stated in Theorem 3.2.  For Case C2 we have ρ − < ρ + ≤ρ. Since ρ + is an unstable asymptote for x → ∞, we must have the constant solution P (x) ≡ ρ + for x ≥ 0. A unique profile P (x) can be obtained by solving this initial value problem backward in x, for x < 0. This unique profile is monotone increasing, see a sample graph in Figure 16. Finally, the profile is not a local attractor for solutions of (1.2), since ρ + is an unstable asymptote. This is further confirmed by the numerical simulation in Figure 17, where we observe oscillation entering the region x > 0 and persisting in time.
For Case C3 withρ ≤ ρ + < ρ − and Case C4 with ρ − >ρ > ρ + , there are no profiles, similar to the results for Case A3 and Case A4. Numerical simulations with Riemann initial data are given in Figure 18 for Case C3, and in Figure 19 for Case C4. In both cases, we observe that oscillations form around x = 0 and enter the region x > 0 and/or x < 0, and the oscillations persist as t grows.
3.2 Case D: κ − < κ + This is the counter part for Case B, and similarly we have 4 sub-cases which we discuss in detail in the following sub sections.
Step 1. On x ≥ 0, the profile can be either the constant function P (x) ≡ ρ + or some horizontal shift of W(x). Unfortunately, the constant function P (x) ≡ ρ + on x ≥ 0 does not work. Indeed, by the connecting condition (3.5) we have As x < 0 get smaller, P (x) grows monotonically, and reaches the value 1 at some finitex, and solution can not continue for x <x. Thus, P (x) is some horizontal shift of W. At x = 0, P (0) takes value between ρ 1 and ρ + , where ρ 1 <ρ < ρ + and f + (ρ 1 ) = f + (ρ + )=f .
Step 2. Consider the initial value problem with P (x) ≡ ρ 1 on x ≥ 0 where ρ 1 is defined in Step 1. By a similar argument as in Step 4 of the proof for Theorem 2.5, the solution is monotone decreasing and approaches ρ − as x → −∞. We call this profile P (x). By the ordering property, all profiles P (x) lie above P (x), therefore P (x) serves as a lower envelope.
Step 3. Let P (x) be a profile such that P (x) equals some horizontal shift of W(x) on x ≥ 0. As P (0) varies from ρ 1 to ρ + , the profile might lose monotonicity and becomes oscillatory. By a similar argument as in Step 5 of the proof for Theorem 2.5, we have, for x < −h, [v(P (x + s)) − v(P (x))]w (s) ds.    Thus, if V (x ) = 0 for some x , then on x ∈ [x , x + h] we have either P (x) ≡ P (x ), or P (x) oscillates around P (x ). Following the same argument as in the rest of Step 5 of the proof for Theorem 2.5, we conclude that there exists an upper envelope P (x) for all the profiles that approaches ρ − as x → −∞. By continuity and the ordering of the profiles, there exist infinitely many profiles between P (x) and P (x).
Sample profiles are given in Figure 20. Using a similar argument as for Theorem 2.3 for Case A1, these profiles are time asymptotic limits for the solutions of the the Cauchy problem of (1.2). We omit the details of the proof. A numerical simulation is presented in Figure 21, with Riemann data, where we observe this asymptotic behavior as t grows.

Case D2, Case D3, and Case D4
For Case D2 we have ρ + < ρ − ≤ρ. Since ρ + is an unstable asymptote for x → ∞, we must have with P (x) ≡ ρ + on x ≥ 0. The unique profile P (x) is constructed by solving this initial value problem backward in x on x < 0. The profile is monotone decreasing, see a sample plot in Figure 22. However, the profile is not an attractor for solutions of the conservation law (1.2), as also indicated by the numerical simulation in Figure 23. We observe that oscillation forms around x = 0, then enters the region x > 0, and it persists in time.
For Case D3 we haveρ ≤ ρ − < ρ + , while for Case D4 we have ρ + <ρ < ρ − . For both cases, there are no profiles, similar to the results in Case B3 and Case B4. Numerical simulation results are presented in Figure 24 for Case D3 and in Figure 25 for Case D4. In both cases, we observe oscillations that enter the region x < 0 and/or x > 0, and they persist in time.

Concluding remarks
We study traveling wave profiles for two nonlocal PDE models for traffic flow with rough road condition. For all possible cases, we show that there can exist infinitely many traveling wave profiles, a unique profile, or no profiles at all, depending on the jump in speed limit and the limits ρ − , ρ + . The stability of these profiles also vary from case to case. Formally, the non-local PDE models are the macroscopic limits of the corresponding nonlocal particle models, referred to as the follow-the-leaders (FtLs) model. In the case where κ(x) ≡ 1, the existence of traveling waves and their convergence to the traveling waves for the PDE models were provided in [15]. It would be interesting to establish a similar result for the FtLs model where κ(x) has a jump. Details will come in a forthcoming work. Codes for all the numerical simulations in this paper can be found at: http://www.personal.psu.edu/wxs27/TrafficNLRough/